Tensor Field Visualization using Fiber Surfaces of Invariant Space

Abstract

Scientific visualization developed successful methods for scalar and vector fields. For tensor fields, however, effective, interactive visualizations are still missing despite progress over the last decades. We present a general approach for the generation of separating surfaces in symmetric, second-order, three-dimensional tensor fields. These surfaces are defined as fiber surfaces of the invariant space, i.e. as pre-images of surfaces in the range of a complete set of invariants. This approach leads to a generalization of the fiber surface algorithm by Klacansky et al. [16] to three dimensions in the range. This is due to the fact that the invariant space is three-dimensional for symmetric second-order tensors over a spatial domain. We present an algorithm for surface construction for simplicial grids in the domain and simplicial surfaces in the invariant space. We demonstrate our approach by applying it to stress fields from component design in mechanical engineering.

Fiber Surfaces for many Variables

“…In this article, we generalized the fiber surface extraction algorithm by Raith et. al [RBN∗19] from 3 to d dimensions in the attribute space to extend the concept of isosurfaces even further. To define a generalized fiber surface, we use a convex ( d – 1)‐dimensional hypersurface.…”

“…We have to intersect a plane with a tetrahedron in three dimensions. It should be noted that we could adapt the method by Raith et al [RBN∗19] in this case. Nevertheless, we operate slightly differently as our hypersurface is defined in another way as we are using hyperplanes instead of triangles.…”

“…Instead, we let users select regions of interest in several three‐dimensional subspaces of the range. In each space, we offer the same solids as interactors as Raith et al [RBN∗19] in their work, i.e., tetrahedra, cubes, triangulated balls, regular cylinders, pyramids, and prisms. All other usual primitives of constructive solid geometry (CSG) could also be used as interactors because they are all convex.…”

“…Basically, this means that we do piecewise linear interpolation on our tetrahedral grid. For further information about the mathematical definition and the feasibility of this mapping, we refer to the paper of Raith et al [RBN∗19]. The next step is to create an interactor inside the current subspace and use it to select a region of interest.…”

“…6. Regarding the used visualization techniques and the discussion of their usefulness in Raith et al [RBN∗19], it would also be possible to visualize the attribute spaces using these techniques as the the algorithm is implemented inside the same framework. Nevertheless, in our opinion the wire‐frame visualization is currently the best one, as sharp features can be seen inside the attribute space which were blurred by a density‐based volume rendering or a related visualization in the spirit of continuous scatterplots.…”